Chemomechanical Origin of Hydrogen Trapping at Grain ... · Chemomechanical Origin of Hydrogen Trapping at Grain Boundaries in fcc Metals Xiao Zhou,1 Daniel Marchand,1 David L. McDowell,2

Chemomechanical Origin of Hydrogen Trapping at Grain Boundaries in fcc Metals

Xiao Zhou,1 Daniel Marchand,1 David L. McDowell,2 Ting Zhu,2 and Jun Song1,*1Department of Mining and Materials Engineering, McGill University, Montréal, Québec H3A 0C5, Canada2Woodruff School of Mechanical Engineering, Georgia Institute of Technology, Atlanta, Georgia 30332, USA

(Received 2 November 2015; published 19 February 2016)

Hydrogen embrittlement of metals is widely observed, but its atomistic origins remain little understoodand much debated. Combining a unique identification of interstitial sites through polyhedral tessellationand first-principles calculations, we study hydrogen adsorption at grain boundaries in a variety of face-centered cubic metals of Ni, Cu, γ-Fe, and Pd. We discover the chemomechanical origin of the variation ofadsorption energetics for interstitial hydrogen at grain boundaries. A general chemomechanical formula isestablished to provide accurate assessments of hydrogen trapping and segregation energetics at grainboundaries, and it also offers direct explanations for certain experimental observations. The present studydeepens our mechanistic understanding of the role of grain boundaries in hydrogen embrittlement andpoints to a viable path towards predictive microstructure engineering against hydrogen embrittlement instructural metals.

DOI: 10.1103/PhysRevLett.116.075502

Despite drastic technological advances in the develop-ment of polymers and composites in thepast several decades,metals remain the irreplaceable backbone inmany importantapplications for the automotive, aerospace, and energyindustries. However, metals are typically susceptible toenvironmental attack. One prominent example is hydrogenembrittlement (HE) that can result in sudden and cata-strophic failure of metallic components and systems [1].Hydrogen is abundant in service environments and manu-facturing processes. As a result, HEposes a significant threatto load-bearingmetallic components and is often consideredas a major obstacle to the reliable applications of structuralmetals.Despite considerable effort in the studyofHE[2–15],the dominant physical mechanisms of HE remain contro-versial [16,17]. Hence, the study of HE at the atomistic andelectronic levelsmay illuminate themechanistic originofHEand thus enable the development of an effective means tomitigate HE. Moreover, the influence of hydrogen ondislocation migration can be distinguished from its role ingrain boundary (GB) segregation and compromise of frac-ture resistance. We focus on the latter here.Hydrogen adsorption is favored at microstructural

heterogeneities [18], such as GBs, as opposed to ininterstitial sites in the bulk lattice. Recently, Bechtle et al.[19] conducted experiments on GB-engineered Ni sampleswith and without hydrogen, and their results showed thatthe susceptibility of HE can be drastically redued at specialGBs that are characterized with low excess free volumesand a high degree of atomic matching. In addition, Oudrisset al. [20] showed that special GBs can trap hydrogen andreduce hydrogen diffusion. These studies highlight theimportant role of GBs in influencing hydrogen transportand embrittlement behaviors and suggest the possibility ofcontrolling the susceptibility of structural metals to HEthrough GB engineering.

To advance rational GB engineering, it is essential tocharacterize the GBs and associated hydrogen segregationbehaviors in a systematic and quantitative manner. Thestructure of certain high angle tilt GBs is commonlydescribed by the coincidence site lattice (CSL) model[21]. Alternatively, the GB structure can be representedby a periodic array of nested three-dimensional (3D)structural units [22,23], which are associated with CSLboundaries but also pertain to general high angle tilt grainboundaries. Along this line of approach, Ashby, Spaepen,and Williams [24] showed that there exist eight uniqueconvex polyhedrons with triangle faces (i.e., so-calleddeltahedra) to account for all possible basic packing unitsat a tilt GB. This approach enables the characterization ofGBs with a simple, yet powerful, concept of a geometricpacking unit, which can be applied to investigate manystructural and chemomechanical properties of GBs, such asinterstitial impurity segregation at GBs [25,26].Here we develop a novel modeling approach that

combines the space tessellation of polyhedral packingunits and first-principles density functional theory (DFT)calculations for studying the hydrogen segregation at GBsin structural metals. Using several face-centered cubic (fcc)metals, including Ni, Cu, γ-Fe, and Pd as representativesystems, we demonstrate that the polyhedral packing unitsat GBs can be uniquely identified and their central holes areshown to serve as favorable interstitial sites of hydrogenadsorption. Our DFT calculations reveal a universaldependence of hydrogen adsorption energies on the localvolume deformation of polyhedral packing units in all fourfcc metals studied. To uncover its physical origin, weestablish a general formula involving a minimum numberof first-principles input and fitting parameters that closelymatch all DFT data of hydrogen adsorption energies atGBs in four different fcc metals. Such a general result

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illuminates the chemomechanical origin of hydrogen seg-regation at GBs. The physical meaning of the parameters inthe formula is clarified. Our results thus provide mecha-nistic insights towards predictive GB engineering to sup-port the development of HE-resistant metals.We have studied a number of symmetric tilt GBs with

various misorientations in the fcc metals of Ni, Cu, γ-Fe,and Pd. Among these GBs, five types of polyhedrons areinvolved: i.e., tetrahedron (TET), octahedron (OCT), pen-tagonal bipyramid (PBP), cap trigonal prism (CTP), andbitetrahedron (BTE) [24], as illustrated in Fig. 1 for arepresentative Σ5ð130Þ½100� GB [meaning the (130) GBface with a [100] tilt axis] in Ni. We identified thesepolyhedrons by space tessellation [27]. Energy minimiza-tion from DFT calculations indicates that there is only oneinterstitial site of hydrogen adsorption in each polyhedron,which lies close to the centroid of the polyhedron. This isconsistent with the Switendick criterion based on theminimum H-H distance [49]. Hence, the center of eachpolyhedron corresponds to an individual hydrogen adsorp-tion site at the GBs. In other words, one can identifypotential hydrogen adsorption sites along GBs using ageometric approach of space tessellation of polyhedralpacking units without a detailed knowledge of hydrogenadsorption chemistry.After the identification of hydrogen adsorption sites at

GBs through space tessellation of polyhedral packing units,we performed DFT calculations to evaluate the interactionsbetween hydrogen and GBs in terms of adsorption ener-getics. Figure 2 shows the differential charge density ofΣ5ð130Þ½100�Ni GB projected along the (100) plane (seeFig. S3 for similar plots of other fcc metals studied). Theseresults indicate that the interactions between hydrogen andthe host atoms are dominantly localized at GBs, and hencethe adsorption energy can be primarily determined by thelocal environment of the capsule, i.e., the polyhedronenclosing the hydrogen atom.

The adsorption energy of hydrogen, Ead, is defined as

Ead ¼ EGBH − EGB − EH

iso ; ð1Þ

where EGB and EGBH are the total energies of the system

before and after adsorption of one hydrogen atom, respec-tively, and EH

iso is the energy of one isolated hydrogen atomin vacuum. Based on Eq. (1), we calculated the hydrogenadsorption energies for different polyhedral sites for aseries of symmetric tilt GBs [27] in Ni, Cu,γ-Fe, and Pd, asplotted in Fig. 3. Clearly, the hydrogen adsorption energiesEad depend largely on the type of polyhedral interstitial site;i.e., the average values of Ead differ for different types ofpolyhedra. Moreover, for each type of polyhedral inter-stitial site, the values of Ead vary markedly. Hence, thepolyhedron is not sufficient alone to uniquely determine theinteractions between hydrogen and GBs.To understand the large variation of Ead, we examined

the local deformation of polyhedrons at GBs. A parameter,dVp=V0

p, is used to measure the local volume changes(dilatation) of polyhedrons. Here, V0

p is the volume ofthe pristine polyhedron, which is defined as the corre-sponding deltahedron with the edge length being thenearest-neighboring distance,

ffiffiffi

2p

a0=2, where a0 denotesthe equilibrium lattice constant in the bulk fcc lattice;furthermore, dVp ¼ Vp − V0

p measures the deviation of theactual polyhedron volume Vp from V0

p. We plot dVp=V0p

together with Ead in Fig. 3. A clear correspondencebetween the variations in the Ead and dVp=V0

p data canbe observed for all four fcc metals studied.The intriguing correspondence shown in Fig. 3 suggests

that the variation of Ead is dictated by the local volumechanges of polyhedral packing units. To elucidate thephysical origin of such a correspondence, we note thatthe mechanical interaction energy between a GB and aninterstitial point defect can be determined by evaluating the

FIG. 1. Schematic illustration of polyhedron in representativeP

5ð130Þ½100�GB and bulk lattice. The gray ball in the dis-tinguished polyhedron represents the host Ni atom, and the smallpink ball in the center is the hydrogen atom.

FIG. 2. Differential charge density of Σ5ð130Þ½100�Ni GBprojected on the (100) plane, with (a)–(e) being two-dimensionalwhile (f)–(j) being three-dimensional charge density contours. In(a)–(e), the blue dashed line represents electron depletion, and thered solid line signifies electron accumulation. In (f)–(j), the redspheres represent Ni atoms locating at the vertices of thepolyhedron that encloses the hydrogen atom, indicated by thesmall pink sphere, while the gray spheres represent other Niatoms. The blue region represents electron depletion, while theyellow region signifies electron accumulation.

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work of PΩp, which corresponds to the local pressure P inthe absence of point defects times the lattice expansion (i.e.,partial volume) Ωp occurring due to the interstitial insertionof a point defect [10,50]. For the present case of hydrogenadsorption at the polyhedral interstitial site, the partialvolume Ωp should be the volume change associated withhydrogen adsorption at the polyhedral interstitial site, andthe local pressure P is determined by the volume changedVp=V0

p according to

P ¼ −BdVp

V0p; ð2Þ

where B is the bulk modulus (see Table I). We note that thebulk modulus of the lattice is an approximation of the localbulk modulus pertaining to GB regions, owing to thedifferences in atomic coordination and bond lengths, butis used here as a first-order approximation. Consequently,the mechanical interaction energy between the hydrogenand polyhedral packing unit is estimated as

dEad ¼ −BΩpdVp

V0p

: ð3Þ

Using Eq. (3), we can express the adsorption energy of ahydrogen atom in a polyhedron packing unit at a GB interms of

Ead ¼ Ead0 − BΩp

dVp

V0p; ð4Þ

where Ead0 is the chemisorption energy of hydrogen in a

deltahedron (i.e., a pristine polyhedron as defined earlier).Equation (4) explicitly reveals the dependence of hydrogenadsorption on the chemisorption energy and the mechanicalinteraction energy, the latter of which is governed by thepartial volume of hydrogen insertion and the local volumedeformation of the polyhedral structural unit at GBs. InFig. 4, the fitting curves (dashed lines) based on Eq. (4)overall agree very well with the data points of adsorptionenergies from DFT calculations. This good agreement alsodirectly explains the correspondence between the variationsin the Ead and dVp=V0

p data as observed in Fig. 3. Thereare, however, noticeable deviations between some of thefirst-principles data and model predictions in the case ofγ-Fe. Such deviations are primarily attributed to ourassumption of the isotropic volumetric deformation ofpolyhedrons in the model [cf. Eq. (4)] while the actualpolyhedron distortion can be anisotropic [27]. This sug-gests the need of a further study to investigate the effect ofdeformation anisotropy of polyhedrons for improving themodel, which will be pursued in the future.Here for simplicity, yet without loss of generality, we

take a single value of partial volumeΩp for all polyhedrons,given an fcc metal studied [51]. The fitting parameters ofEad0 and Ωp. are listed in Table I [27].The close agreement between the DFT data and the

predictions based on Eq. (4) shown in Fig. 4 demonstratesthat Eq. (4) captures the dominant chemomechanical effectsof hydrogen adsorption and segregation at GBs. In Eq. (4),Ead0 can be regarded as an intrinsic property of pristine

FIG. 3. The variation in the hydrogen adsorption energy Ead

and normalized lattice dilatation dVp=dV0p of polyhedrons in (a)

Ni, (b) Cu, (c) γ-Fe, and (d) Pd systems.

TABLE I. List of material properties, i.e., bulk modulus (B), bulk hydrogen partial volume (Ω), predicted hydrogen partial volume atpolyhedrons (Ωp), and model predicted (and DFT calculated) adsorption energy of hydrogen in a pristine polyhedron, i.e.,½Model�Ead

0 ð½DFT�Ead0 Þ, in examined material systems.

System

Properties Ni Cu γ-Fe Pd

B (GPa) 195 137 281 168Ω (Å3) 2.28 2.68 2.07 2.42Ωp (Å3) 2.03 2.54 1.76 2.19½Model�Ead

0 ð½DFT�Ead0 Þ (eV) TET −2.11 (−2.06) −1.72 (−1.72) −1.82 (−1.87) −2.38 (−2.35)

OCT −2.24 (−2.26) −1.86 (−1.85) −2.14 (−2.18) −2.40 (−2.38)BTE −2.05 (−2.04) −1.73 (−1.70) −1.83 (−1.79) −2.27 (−2.30)PBP −1.85 −1.59 −1.79 −2.06CTP −2.20 −1.75 −2.04 −2.18

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polyhedrons associated with the chemisorption of hydro-gen, which can be separately determined (other than the GBcalculations). The pristine TETs and OCTs are commonlypresent in a bulk fcc lattice, the pristine BTEs are basicconstituents of coherent twin boundaries, and the corre-sponding hydrogen adsorption energies can be readilyevaluated. These data, also listed in Table I, are in closeagreement with the Ead

0 values obtained from the previousGB calculations. This validates the treatment of Ead

0 as amaterial constant. Incidentally, the pristine PBP and OCTpolyhedrons are not present in the GB structures examinedin the present study. Nonetheless, we suggest the possiblemethods (elaborated in details in Ref. [27]) by which onemight construct pseudopristine PBP and OCT polyhedronsto compute the corresponding values of Ead

0 . In addition, wenote that, in Table I, OCT (the polyhedron responsible forH adsorption in bulk lattice) exhibits the lowest Ead

0 amongall five polyhedrons. Besides, the partial volume of hydro-gen adsorption at GBs, Ωp, is another important parameterin Eq. (4). Interestingly, the value of Ωp obtained is nearlyidentical to the partial volume of a hydrogen interstitial inthe bulk lattice (see Table I). Hence, hydrogen induces asimilar dilatation both at the GB and in the bulk [27].Equation (4) provides a predictive model for evaluating

the energetics of hydrogen trapping and segregation atGBs. It is important to point out that this model is a verygeneral, physics-based model. Besides the fcc metal sys-tems, the model is also expected to be applicable to metalsof other crystal structures (e.g., hcp and bcc). Somepreliminary calculations have been performed using bccNb as an example to demonstrate the generality of themodel [27]. In light of recent experiments [19,20,52,53] onhydrogen embrittlement, several case studies of hydrogenembrittlement of GBs were performed, as elaborated inRef. [27]. Equation (4) enables a quick evaluation of thetendency of hydrogen segregation at GBs. For instance, the

Σ3 and Σ3n families (i.e., the “special” GBs defined inRef. [19] and discussed earlier in this paper) and Σ11 GBexhibit a lack of volume changes of polyhedral structuralunits. As a result, they are unfavorable to hydrogentrapping and thus less prone to hydrogen embrittlementin terms of less decrease of work of separation of GBs. Incontrast, other GBs with high sigma numbers, such as Σ17and Σ73, involve substantial volume changes of polyhedralstructural units and are more susceptible to hydrogentrapping and accumulation, thus giving rise to more severeembrittlement effects (see Figs. S8 and S9 in Ref. [27]).This trend is consistent with the experimental observationsof hydrogen embrittlement effects on GB-engineered Ni[19], where Ni samples consisting of high-density specialboundaries (i.e., principally Σ3 twin boundaries) aredemonstrated to exhibit good HE resistance.Moreover, our study suggests a mechanistic pathway for

further study of the GB effects on hydrogen embrittlement.First, with polyhedrons as atomic structural units of the metallattice, the diffusion of hydrogen can be considered asdiscrete hops between neighboring polyhedra. Given thehighly localized interactions between hydrogen and GBs, thejumping trial frequency and migration barrier would pre-sumably depend on the coupling of neighboring polyhedraand their associated dilatation. Recognition of such localizedinteractions will enable the characterization of the completediffusion parameters through a finite set of calculations [54],thus greatly facilitating the study of the kinetics of hydrogenmigration at GBs. Second, the present study calls for arigorous continuum micromechanical study on the deforma-tion fields of GBs, e.g., through the generalized Peierls-Nabarro model that treats the atomic interaction right at theGB interface and the continuum elastic interaction for therest of system [55,56]. This will enable the prediction ofvolume distortion, dVp=V0

p at GBs directly from continuummicromechanics, and thus reduce the need of intensive first-principles calculations, besides the intrinsic properties ofpolyhedrons, such as Ead

0 and Ωp. As such, Eq. (4), inconjunction with the aforementioned analyses, would pro-vide a full-scale predictive framework to quantitatively guidethe GB engineering against hydrogen embrittlement.In summary, we study the energetics of hydrogen

adsorption for a variety of GB structures by combiningthe space tessellation of polyhedral packing units and thefirst-principles calculations. We further develop a physics-based, predictive model, as given by Eq. (4), to reveal thechemomechanical origin of hydrogen trapping and segre-gation at GBs. This model is validated through thequantitative evaluation of hydrogen adsorption energiesas a function of volumetric deformation of polyhedralstructural units at GBs for several fcc metals. Our resultsadvance the atomic-level understanding of the role of GBsin hydrogen embrittlement and provide mechanisticinsights that may enable predictive GB engineering againsthydrogen embrittlement. Such insights can be also fed into

FIG. 4. Comparison between the DFT calculated (open sym-bols) and model predicted (dashed lines) adsorption energeticsversus the volume distortion relation in (a) Ni, (b) Cu, (c) -Fe, and(d) Pd systems.

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the study of hydrogen adsorption kinetics and fracturemechanics for advancing our understanding of hydrogen-assisted GB decohesion or cracking.

J. S. acknowledges the financial support from McGillEngineering Doctoral Award and National Sciences andEngineering Research Council (NSERC) Discovery grant(Grant No. RGPIN 418469-2012). D. L. M. and T. Z.acknowledge support from QuesTek to study hydrogeneffects in metals. X. Z. acknowledges the financial supportfrom China Scholarship Council (CSC). We also acknowl-edge Supercomputer Consortium Laval UQAMMcGill andEastern Quebec for providing computing power.

*To whom all correspondence should be [email protected]

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